Asymptotic behavior of connecting-nearest-neighbor models for growing networks

This paper deals with the mathematical description of the asymptotic behavior of the solutions of a couple of models for the dynamics of growing networks based on connecting, with a higher probability, nodes that have a neighbor in common. The first model, proposed by A. Vázquez, is nonlinear and, in general, the long-time behavior of the solutions differs from the one predicted by the linear reduction proposed in its original treatment. A second model is specifically derived from the rules defining an in silico model also proposed by Vázquez to simulate the growth of a network under the mechanism of connecting nearest neighbors. The two analytical models lead to very different predictions for the configuration of the network that are tested using the simulations of the in silico model.